Introduction to Probability and Statistics for Engineers and Scientists

*9.10Multiple Linear Regression 403

=(Y′−B′X′)(Y−XB)

=Y′Y−Y′XB−B′X′Y+B′X′XB

=Y′Y−Y′XB

where the last equality follows from the normal equations

X′XB=X′Y

BecauseY′is 1×n,Xisn×p, andBisp×1, it follows thatY′XBisa1×1 matrix.
That is,Y′XBis a scalar and thus is equal to its transpose, which shows that

Y′XB=(Y′XB)′

=B′X′Y

Hence, using Equation 9.10.7 we have proven the following identity:

SSR=Y′Y−B′X′Y

The foregoing is a useful computational formula forSSR(though one must be careful
of possible roundoff error when using it).

EXAMPLE 9.10b For the data of Example 9.10a, we computed thatSSR=34.12. Since
n=8,k=2, the estimate ofσ^2 is 34.12/5=6.824. ■

EXAMPLE 9.10c The diameter of a tree at its breast height is influenced by many factors.
The data in Table 9.5 relate the diameter of a particular type of eucalyptus tree to its age,
average rainfall at its site, site’s elevation, and the wood’s mean specific gravity. (The data
come from R. G. Skolmen, 1975, “Shrinkage and Specific Gravity Variation in Robusta
Eucalyptus Wood Grown in Hawaii.” USDA Forest Service PSW-298.)
Assuming a linear regression model of the form

Y=β 0 +β 1 x 1 +β 2 x 2 +β 3 x 3 +β 4 x 4 +e

wherex 1 is the age,x 2 is the elevation,x 3 is the rainfall,x 4 is the specific gravity, andYis
the tree’s diameter, test the hypothesis thatβ 2 =0. That is, test the hypothesis that, given
the other three factors, the elevation of the tree does not affect its diameter.

SOLUTION Totestthishypothesis, webeginbyrunningProgram9.10, whichyields, among
other things, the following:

(X′X)−3,3^1 =.379, SSR=19.262, B 2 =.075